Revisiting nucleosynthesis constraints on primordial magnetic fields

27June1996

PHYSICS

LETTERS B

Physics Letters B 379 (1996) 73-79

ELSEVIER

Revisiting nucleosynthesis constraints on primordial magnetic fields Dario Grass0 ‘, Hector R. Rubinstein Department of Theoretical Physics, Uppsala University, Box 803, S-751 08 Vppsala, Sweden and Department of Physics, University of Stockholm, Vanadisviigen 9, S-113 46 Stockholm, Sweden

Received 15 February 1996 Editor: P.V. Landshoff

Abstract In view of several conflicting results, we reanalyze the effects of magnetic fields on the primordial nucleosynthesis. In the case the magnetic field is homogeneous over a horizon volume, we show that the main effects of the magnetic field are given by the contribution of its energy density to the Universe expansion rate and the effect of the field on the electrons quantum statistics. Although, in order to get an upper limit on the field strength, the weight of the former effect is numerically larger, the latter cannot be neglected. Including both effects in the PN code we get the upper limit B 2 1 x 10” Gauss at the temperature T = lo9 K. We generalize the considerations to cases when instead the magnetic is inhomogeneous on the horizon length. We show that in these cases only the effect of the magnetic field on the electron statistics is relevant. If the coherence length of the magnetic field at the end of the PN is in the range 10 < Z,IJ< IO” cm our upper limit is B 5 1 x IO'* Gauss. PACS: 98.8O.Cq; 98.62.En. Keywords: Primordial nucleosynthesis; Primordial magnetic fields

The study of the effects of magnetic fields on the primordial nucleosynthesis (PN) began with the pioneering works of Greenstein [ 1] and Matese and O’Connell [2]. Matese and O’Connell in a previous paper [3] first observed that magnetic fields larger than the critical value B, E eB/ma = 4.4 x 1013 Gauss can affect the P-decay rate, mainly through the modification of the phase space of the electrons and positrons. Greenstein advanced that, if magnetic fields were present in the early Universe, then this can affect not only the weak processes regulating

’ Supported by a EEC Twinning grant.

the neutron namely

to proton

ratio in the early Universe,

12 + e+ +p+F n + v *

p + e-

n+p+e-+F but also the expansion rate of the Universe. These effects will have consequences on the PN. However, they are competing effects. In fact, whereas the magnetic field energy density PB = B2/89r accelerates the Universe expansion and therefore increases the predicted abundance of the relic 4He, the effect of the

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74

D. Grasso, H.R. Rubinstein/Physics LettersB 379 (1996) 73-79

field on the weak processes is to increase the n -+ p conversion rate reducing the predicted 4He abundance. Greenstein argued that the former effect dominates over the latter. In a following paper, Matese and O’Connell [ 21 improved Greenstein’s considerations by computing explicitly n --) p conversion rate at the PN time in presence of a strong magnetic field and they confirmed Greenstein’s qualitative conclusions. The comparison of the PN predictions for the relic light isotope abundances with the observations provides a criteria to bound the magnetic field strength at the nucleosynthesis time [ 41. However, Matese and O’Connell did not provide any upper limit. Recently, the interest about magnetic fields in the early Universe received new impulse from the idea that these fields might be produced during the electroweak phase transition [5] or inflation [ 61. The renewed interest in this subject led several authors to investigate Matese and O’Connell’s treatment again. Cheng, Schramm and Truran [7] first used the standard nucleosynthesis code [ 81 to get an upper limit on the magnetic field strength. The effects of the field they considered are the same studied by Matese and O’Connell. However Cheng et al. disagree with Matese and O’Connell in their conclusions. In fact, they claim that the effect of the magnetic field on the weak reactions is the most important. Cheng et al. obtained the following upper limit: B < 10” Gauss at the end of the nucleosynthesis (T = 10’ K) . In both the Matese and O’Connell and Cheng et al. papers the magnetic field was assumed to be uniform over an horizon volume. We will return below to the crucial implications of this assumption. Recently, we reconsidered this subject [ 91. Besides the effects of the magnetic field on the reaction rates, we considered the effect of the field on the statistical distribution of the electrons and positrons. In fact, due to the effect of the magnetic field on the electron wave function, the phase space of the lowest Landau level is enhanced. If the temperature is small with respect to the energy gap between the lowest and first exited level, that is if eB > T2 2, a relevant fraction of electron-positron pairs will condens’e in the lowest Landau level affecting the statistical distribution of the eiectron gas. As a consequence, the number * Observe that, neglecting dissipation, during the Universe expansion.

eB/T* remains constant

density and energy density of electrons and positrons are increased with respect to the case where the magnetic field is not present. The expressions for the electron+positron energy density and pressure are y nz4-52 pe=s ena

-

6”())

co

J

X

2

de

$--

1-2(n+

l)r

&?izEJ X

pe=

1

(

1

Ifey+P+

l+ey-"

Y 40° 671_2me c“=0 (2 -

> '

&lo>

00

X

J

de

(2-

1) 8

E2 - 1 - 2(n + l>y

&5z5 X

1 lfey+P

1 +

l+ey-F

(2) >

Here y E B/B,, E and ,u are the electron, or positron, energy and chemical potential in mass units, and n labels the Landau level. It is evident that the modified contribution of electrons and positrons to the total energy density will affect the expansion rate of the Universe. The entropy content of the Universe will be also changed by the effect on the field on the electrons thermodynamics. However, for values of the ratio eB/T2 N 1, the main effect of the modified electron statistics on PN is on the variation that it induces on the time derivative of the photon temperature. This derivative is given by

where pern = pe + pr, pem = pe + pr, KT is the Universe expansion rate and T is the photon temperature. For small values of the ratio eB/T2, the most relevant effect of the magnetic field enters in the derivative dp,/dT? that is smaller than the value it would have if the field were not present. More physically, this effect can be interpreted as a delay in the electronpositron annihilation time induced by the magnetic field. This will give rise to a slower entropy transfer

D. Grasso, H.R. Rubinstein /Physics Letters B 379 (1996) 73-79

from the electron-positron pairs to the photons, then to a slower reheating of the heat bath. In fact, due to the enlarged phase-space of the lowest Landau level of electrons and positrons, the equilibrium of the process e+e- ++ y is shifted towards its left side. As we already showed in Ref. [ 93, and we are going to discuss in more details here, this effect has a clear signature on the Deuterium and 3He predicted abundances besides that on the 4He abundance. Including the effect of the magnetic field on the electron statistics, together with the effect of the field energy density and the effect on the weak processes in the standard nucleosynthesis code 3 we got the upper limit B 5 3 x lOLo Gauss. We also considered other kind of effects like that of the magnetic field on the nucleon [ IO] and electron masses, but these effects proved to be subdominant. Recently our conclusions have been put in question in a preprint by Kernan, Starkman and Vachaspati [ Ill. In this preprint, the effect of the magnetic field on the PN has been parameterized in terms of an effective number of neutrinos in order to avoid the numerical PN computations. Kernan et al. considered, at least in principle, all the effects of the magnetic field that we discussed above, including that on the electron statistics. However, although for opposite reasons, their main conclusion disagree both with Cheng et al. and our results. In fact, in Kernan et al. opinion the most.relevant consequence of the magnetic field on the PN is the one mediated by the effect of the field energy density on the expansion rate of the Universe. The upper limit on the strength of the magnetic field they got is eB/Tz 2 1 that can be read B 5 1 x 10’ 1 at T = log K. This result does not differ from Cheng et al. [7] numerical result although it does in its physical interpretation. The aim of this letter is to try to clarify this confused situation. As we are going to show, some mistakes are probably present in all Refs. [ 7,11,9]. To start with, we have recently realized that our upper limit, in Ref. [9] is incorrect. This mistake was due to the use of a wrong version of the standard nucleosynthesis code. We have now rerun the right version of the PN code [ 121 with an improved subroutine that evaluates the effect of the magnetic field on the electron and positron energy density, pressure and energy density 3 However we used here an old version of the code, see [ 81.

7.5

Table 1 Here we report the PN predictions for the relic light element abundance for several values of the parameter y s B/B= given at the temperature T = lo9 K y(T=

1 2 3 4 5 6 7 8 9 1

x x x x x x x x x x

lo9 K)

4He

(D+3He)/H

‘Li/H

0 10-3 10-S 10-X 10-3 10-3 1O-3 10-3 1O-3 10-3 10-J

0.237 0.240 0.242 0.244 0.246 0.249 0.252 0.255 0.258 0.262 0.266

1.04 1.02 1.02 1.03 1.05 1.08 1.12 1.16 1.22 1.28 1.35

1.15 1.17 1.19 1.19 1.20 1.21 1.22 1.24 1.27 1.31 1.36

x x x x x x x x x x x

10-4 10-h 10-4 10-d 10-d 1O-4 10-4 1O-4 10-d 1O-4 10-4

x x x x x x x x x x x

lo-” lo-‘0 lo-‘0 lo-*0 10-10 10-10 10-10 lo-lo lo-lo lo-‘0 lo-I0

time derivative. We considered also the effects of the magnetic field on the weak reaction and the effect of the field energy density on the Universe expansion rate. Including all possible effects in the PN code we get the predictions we report in Table 1. These results have been obtained using N,, = 3, the lowest value of the neutron half life compatible with experimental data (71/2(n) = 885 s.), and the smallest value of the baryons to photons ratio that makes the predicted (D +3 He) /H ratio, in absence of the magnetic field, compatible with the astrophysical observations (v = 2.8 x lo-“). This choice assured the minimal predicted abundance of the 4He for each value of the magnetic field strength that we considered. Requiring that such abundance do not exceed the value of 0.245 [ 131 we infer from our data the upper limit B 5 1 x 10” Gauss

(4)

when the temperature was T = 10’ K (end of PN) . In order to discriminate the relative weight of the effects we considered we have run our code leaving on only one effect at a time. We start considering only the effect of the magnetic field energy density. As expected, we see in Table 2 that the predicted 4He abundance grows with the field strength. In fact, larger values of B induce a larger Universe expansion rate then a higher freeze-out temperature for the n/p ratio. In Table 3 we report instead our predictions in the case only the effect of the magnetic field on the electron quantum statistic is considered. Note that, in order to display the relative weight of the effects that

D. Grasso, H.R. Rubinstein /Physics Letters B 379 (1996) 73-79

76 Table 2 Here we report

PN predictions obtained considering only the effect of the magnetic field energy density on the Universe the

expansion ~(7. = 10’ K)

1x 2x 3x 4x 5x 6x 7x 8x 9x 1x

0 10-S 10-x lo-” 10-s 10-S IO-” 10-3 lo-’ IO-” 10-2

4He

(D+3He) /H

7Li/H

0.237 0.237 0.238 0.240 0.241 0.244 0.246 0.249 0.252 0.256 0.259

1.04 x 1.05 x 1.06 x 1.09 x 1.12 x 1.17 x 1.22 x 1.28 x 1.36 x 1.44 x 1.53 x

1.15 x 1.15 x 1.15 x 1.15 x 1.16 x 1.17 x 1.19 x 1.23 x 1.28 x 1.34 x 1.42 x

10-4 10-d 1O-4 10-d 10-4 10-4 10-4 lO-4 10-4 10-4 10-4

10-l” lo-‘” lo-‘” 10-l” 10-l” 10-l” 10-l” lo-‘” lo-‘* lo-‘” 10-l”

Table 3 Here we report the PN predictions obtained considering only the effect of the magnetic field energy positrons quantum statistics y(T = IO9 K)

0 1 x IO-3 2 x lo-” 3 x IO-3 4 x 10-j 5 x 10-3 6 x IO-’ 7 x 10-3 8 x IO-” 9 x 10-x 1 x 10-2 1.1 x 10-2 1.2 x 10-2

density

on the electrons

4He

(D-t3He)/H

7Li/H

0.237 0.240 0.241 0.241 0.242 0.243 0.243 0.243 0.244 0.244 0.244 0.245 0.245

1.04 x 1.01 x 9.99 x 9.89 x 9.81 x 9.72 x 9.65 x 9.59 x 9.51 x 9.46 x 9.40 x 9.35 x 9.28 x

1.15 x 1.17 x 1.19 x 1.20 x 1.21 x 1.22 x 1.22 x 1.23 x 1.24 x 1.24 x 1.25 x 1.26 x 1.26 x

10-4 10-d 10-5 lOF5 lo@ lo@ lO-5 10-S 10-5 lO-5 10-S 10-5 lO-5

and

10-l” 10-l” lo-‘” lo-‘” lo-‘” 10-l” 10-l” lo-‘” lo-‘” lo-‘” lo-‘” lo-‘” lo-lo

we considered, deriving the results of Tables 2 and 3 we kept the values of ~-1/2(n) and 7 fixed and equal to the value we used for Table 1. It is interesting to observe the behavior of the sum of the predicted abundances for the Deuterium and 3He while vary’m g the magnetic field strength. This is qualitatively different from what reported in Table 2. In fact, if only the effect on the electron statistics is considered, the (D + 3He) /H abundances ratio decreases when the field strength increases (see also Fig. 2). At the same time, the predicted 4He increases increasing B (see Fig. 1). From the results we reported in Table 2 it follows

that the upper limit we would get if only the effect of the magnetic field energy density were present is B 5 2 x 10” Gauss whereas considering only the effect on the electrons quantum statistics (see Table 3) we wouldgetB 5 5x 10 ” 4 . Thus, contrary to our previous claim [91, the former effect numerically dominates the latter. However, from Fig. 1 it is clear at a glance that the effect of the magnetic field on the electron statistics cannot be neglected and indeed it dominates for small values of the field strength. In this sense we still disagree with the conclusion of the authors of Ref. [ 11 I. It may be that in Ref. [ 1 I ] the effect of the magnetic field on the electrons and positrons statistical distribution was not treated properly close to the saturation of the upper limit on B, that is for eB/T* N 1. Concerning the effect of the magnetic field on the weak reactions we have verified that such effect is negligible with respect to the others. Mass changes are also negligible in the range of field strengths that we considered. About these conclusions we agree with the authors of the Refs. [ 21 and [ 111. Since the numerical predictions of Ref. [ 71 do not differ significantly from those of Ref. [ I I] and the results we reported in Table 2 we think that the different physical interpretation of the results given in Ref. [7] is probably due to an oversight writing that paper. To summarize, we have found that the main effects of a magnetic field on the PN plays through the action of the field on the electrons and positrons quantum statistics and the direct effect of the field energy density on the expansion rate of the Universe. Although the latter effect is numerically larger than the former, these effects are indeed comparable, at least if the magnetic field is uniform over the horizon scale as we assumed so far. Let us now discuss how our previous consideration will be affected if we relax the uniformity assumption. Before discussing more model dependent motivations to consider inhomogeneous magnetic fields, it is mandatory to compare the upper limit we got above with the upper limit one would get considering the effect of an homogeneous cosmic magnetic field 4 We have to point out that this limit has been obtained using only our predictions for the 4He abundance keeping the value of 7 fixed. However, since 7 is an unknown parameter, it should be resealed for every chosen value of B in such a way to get the largest D + 3He abundance compatible with the observational data (see below).

D. Grasso, H.R. Rubinstein/Physics

Fig. 1. The 4He predicted abundance is represented in function of the parameter y, considered at T = IO9 K, in three different cases: only the effect of the magnetic field energy density is considered (dashed line) ; only the effect of the field on the electron statistics is considered (dotted-dashed line j; both effects are considered (continuous line). The dotted line represents the observational upper limit.

Fig. 2. The (D+3He) /H predicted abundance the same cases illustrated in Fig. 1.

is represented

for

the early Universe isotropy. In fact, since magfields break rotational invariance they can induce an anisotropy in the Universe expansion. This anisotropy can have observable consequences on the isotropy of the cosmic background radiation and on the PN [ 14,151. Using these criteria Zeldovich and Novikov [ 141 got the limit B 5 4 x 10” Gauss for a magnetic field homogeneous at least over a horizon volume considered at the temperature T = 10’ K. This limit is roughly one order of magnitude more stringent than the one we got above as those obtained in Refs. [7] and [ll] as well. on

netic

Letters B 379 (1996) 73-79

77

Nevertheless, if the magnetic field is inhomogeneous with a coherence length much smaller than the Hubble radius we do not have to care about the Zeldovich and Novikov upper limit. In our opinion this is the most plausible physical scenario. It is in agreement with the predictions of most of the models for magnetic field generation in the early Universe [ 161. Among these models we are mainly interested in those that predict that magnetic fields are generated during the electroweak phase transition. The predicted coherence length of the field at the electroweak phase transition time is Lo M rnw’ N 10-‘6H,-,’ (where H,-,’ is the Hubble radius at that time) in the case the transition is second order [S] or, perhaps a more probable scenario, La N &bbte x 10-(2+3)H&,,1 for a firstorder phase transition [ 17,181. Assuming the magnetic field strength decreases only due the Universe expansion the ratio b/H-’ remains constant in time. In both cases it is evident that La < H-‘. Having magnetic fields that fluctuate in space on a typical scale Z& < H-t forces us to reconsider the analysis we made in the first part of this letter paying some attention to the different scales on which the effects of the field on the PN act and the nature of the averaging processes. To start with, let consider the effect of the field on the weak processes. The mean free path between two weak reactions is IW N GcT5 that is of the order of the horizon radius at the onset of the PN (H-l (T N 10” K) N 10” cm). It is clear that if & < I;’ the charged particles involved in the process will feel a magnetic field changing a number of times between the two reactions so that the effect of the field will be almost averaged out [ 191. At any rate, the effect of the field on the weak reaction rates was already subdominant in the case B were assumed to be homogeneous. More relevant is, instead, the effect of the inhomogeneity of the magnetic field on the action that the magnetic field energy density PB plays on the Universe expansion. As discussed in detail in Refs. [ 19,201, since in this case the field is fluctuating in space, only a mean magnetic field within a horizon volume can be defined as a meaningful quantity. This quantity is defined by LS

1

d3r B&,,,(r)

ijia = K s

Lo

(5)

78

D. Grasso, H.R. Rubinstein / Physics Letters B 379 (1996) 73-79

where

&ms=Bo

(;)‘($

(6)

is the root-mean-squared field strength. Here p is a parameter that depends on the statistical configuration of the magnetic field. Typical values of p are l/2, 1, 3/2. Inserting Eq. (6) in Eq. (5) it is evident that ps will suffer a huge suppression if LO < LH so that it will have no chances to affect any more the PN. The impact of the magnetic field on the electron quantum statistics is much less affected than the others by the inhomogeneity of the field. In fact, in this case, the characteristic length scale we have to compare to ,?& is the Compton scattering length of the electrons Ac, since this scattering is responsible for the electron thermalization, and the radius of periodic motion of the electrons in the plane normal to the magnetic field vector. At the PN temperature (T N 109’lo K) the Compton cross section does not differ significantly from the Thomson cross section. Then we have Ac = (~rfi~u)-’

N 10-(2’5)

cm.

(7)

This has to be compared with the Hubble radius at the same temperature that is N lO’O+’ ’ cm. The radius of the classical orbit of the electron in an over critical magnetic fields is given by

the relic abundances extrapolated from present observations. Since in our case the predicted of abundances of these isotopes depend on the field strength, we have to rescale 7 for every value of B in order to get their largest predicted abundances compatible with the observational upper limit. According to Ref. [ 131 we assume this limit to be (D +3 He) /H < 1. B x 10e4 . Finally, we have to compare the predicted abundance of the 4He, obtained using such value of r), with the maximal abundance compatible with observations 4He 5 0.245 [ 13 ] . Using these procedure we get the upper limit B 5 1 x 1Or2 Gauss.

(9)

Again, this limit refers to the temperature T = lo9 K. The corresponding value of 77is 2.3. As expected, this value is smaller than the standard lower limit 7 2 2.8, since the effect of the magnetic field on the electron statistics suppresses the predicted abundance of D and 3He. It is worthwhile to note here that, although this effect increases also the predicted abundance of 4He, large magnetic fields might help to alleviate the “nucleosynthesis crisis” that is presently under debate

[211. So far we neglected any dissipative effect in the plasma. However, even in a relativistic plasma conductivity (+ is a finite quantity. This means that any field configuration on a length scale La < L&s, where H-’

where n labels the Landau level. Since the magnetic field plays its main effect of on the lowest Landau level it is clear that the radius we have to care about is much smaller than hc and far below the expected value of LO at the PN time. Since, AC < I,’ < H-l we see that the effect of the magnetic field on the electron quantum statistics provides the best probe of the field at small length scales. If hc << b < H-’ this is the only effect we remain with. The PN predictions of Table 3 cannot be used in their present form to determine, in this case, the upper limit on the magnetic field strength. In fact, as we pointed-out above, obtaining Table 3 we kept the value of v fixed. However, 7 is an unknown parameter and its value has to be bound from below in order the predicted abundances of D and 3He do not exceed

Ldiss

=

(10)

\i= &i-u

will be dissipated. It is interesting to observe that if eB/T2 >> 1 the conductivity properties of the plasma will be seriously affected by the magnetic field. In fact, in this case the e+e- pair number density is increased by the effect of the field on the electrons and positrons phase-space. As a consequence this will make the plasma conductivity to decrease then J&s to grow [ 221. This can have important consequences on the field evolution after it is generated in the early Universe. However, such effect is negligible at the PN time. Following Ref. [20] Lass at T = lo9 K MeV can be estimated to be L&

crl

0.1 g,-1/4

lo9 7’4 M

( T

>

1 +

10

cm.

(11)

D. Grasso, H.R. Rubinstein/Physics

Taking this considerations into account we conclude that our upper limit (9) applies only if 10 < ,Q << 10” cm. It is worthwhile to see how our upper limit constraints at least one of the recent models of relic magnetic field generation. Assuming the field is generated at the end of a first-order electroweak phase-transition [ 17,181, within large theoretical uncertainties the predicted magnetic field strength at the temperature T N I015K is M 1O24Gauss with a coherence length Le M 10--(2+3) cm. If the magnetic field flux above this length scale remains frozen in the plasma, so that the field strength decrease only due to the Universe expansion, at the PN time we expect: B(T = lo9K) N 10” Gauss and Lo N 104”5 cm. Thus, this model is not excluded by PN considerations. In conclusion, in this letter we unfortunately revise upwards previous limits on homogeneous magnetic field strengths at PN time by a factor 10. Earlier calculations contained mistakes on a variety of points as discussed. As a consequence, previous bounds based on Universe isotropy remain more stringent [ 141. However, in the case the magnetic field is inhomogeneous on the horizon length scale, isotropy considerations do not apply. In this case, we showed that the main observable effect of the magnetic field on the PN is on the electron and positron quantum statistics. The authors wish to thank V. Semikoz for valuable discussions. This work was supported in part by the Swedish Research Council and the Twinning EEC contract number SCl*-CT91-0650 (D.G.). References [ I ] G. Greenstein, Nature 233 (1969) 938. ]2] J.J. Matese and RF. O’Connell, Astrophys. J. 160 (1970) 451. 131 J.J. Matese and RF. O’Connell, Phys. Rev. 180 (1969) 1289.

Letters B 379 (1996) 73-79

79

[4] For a review of most aspects of primordial nucleosynthesis big bang cosmology, see E.W. Kolb and MS. Turner, The Early Universe (Addison-Wesley Reading, MA, 1989). [51 T. Vachaspati, Phys. Lett. B 265 (1991) 258. [6] see, e.g.: M. Gasperini, M. Giovannini and G. Veneziano, Phys. Rev. Lett. 75 (1995) 3796; A.C. Davis and K. Dimopoulos, CERN-TH/95, DAMPT95 31, astro-ph/9506132. [7] B. Cheng, D.N. Schramm and J.W. Truran, Phys. Rev. D 49 (1994) 5006; Phys. Lett. B 316 (1993) 521. [S] L. Kawano, Let’s Go Early Universe: Guide to Primordial Nucleosynthesis Programming, Fermilab-PUB-88/34-A. This is a modernized and optimized version of the code written by R.V. Wagoner, Astrophys. J. 179 (1973) 343. [9] D. Grass0 and H.R. Rubinstein, Astrophys. Phys. 3 (1995) 95. [lo] M. Bander and H.R. Rubinstein, Phys. L&t. B 311 (1993) 187. [ll] PJ. Keman,G.D. Starkman and T. Vachaspati, preprint CWRU-P IO-95 astro-ph/9509126. [ 121 L. Kawano, Let’s Go Early Universe (2): Primordial Nucleosynthesis the Computer Way, Fermilab-PUB-92/04-A (January 1992). [ 131 Since the updated relic abundances are still subject of discussion (see Ref. [21] ) we prefer to adopt here a conservative point of view using the abundances reported in: TX Walker, G. Steigman, D.N. Schramm, K.A. Olive and H-S. Kang, Astrophys. J. 376 (1991) 51. [ 141 see Y.B. Zeldovich, A.A. Ruzmakin and D.D. Sokoloff, Magnetic Fields in Astrophysics, p. 293 (Gordon and Breach, London, 1983) and references therein. [15] KS. Thome, Astrophys. J. 148 (1967) 51. [ 161 for a review see PP. Kronberg, Rep. Prog. Phys. 57 (1994) 325. [ 171 T.W.B. Kibble and A. Vilenkin, Phys. Rev. D 52 (1995) 679. [ 181 G. Baym, D. Biideker and L. McLerran, Phys. Rev. D 53 (1996) 662. [ 191 K. Enqvist and P Olesen, Phys. Lett. B 319 ( 1993) 178. [20] K. Enqvist, A.I. Rez and V.B. Semikoz, Nucl. Phys. B 436 (1995) 49. [21] see, e.g.: N. Hata et al., Phys. Rev. Lett. 75 (1995) 3977; C.J. Copi, D.N. Schramm and M.S. Turner, Phys. Rev. Lett. 75 (1995) 3981. [22] U.H. Danielsson and D. Grasso, Phys. Rev. D 52 (1995) 2533.